Let's review the problem presented in the image with the focus on the equation $\cos\theta + 8\sin\theta = 4$.

Given Information

The equation from the image is:

$\cos\theta + 8\sin\theta = 4$

And it is part of a system of equations used to solve for $\theta$ and a variable $x$. The steps provided in the image seem to be using a geometric approach.

Steps for Solving the Equation

1. Transform the Trigonometric Equation: To solve $\cos\theta + 8\sin\theta = 4$, you can rewrite it in the form of a single trigonometric function: $R\cos(\theta - \alpha) = 4$ where $R$ and $\alpha$ are defined by: $R = \sqrt{\cos^2\theta + (8\sin\theta)^2} = \sqrt{1 + 64} = \sqrt{65}$ $\tan\alpha = \frac{8}{1} \implies \alpha = \tan^{-1}(8)$

   Thus the equation can be rewritten as: $\sqrt{65}\cos(\theta - \alpha) = 4$ So: $\cos(\theta - \alpha) = \frac{4}{\sqrt{65}}$

2. Solve for $\theta$: Using the inverse cosine function: $\theta - \alpha = \cos^{-1}\left(\frac{4}{\sqrt{65}}\right)$ Then solve for $\theta$: $\theta = \alpha + \cos^{-1}\left(\frac{4}{\sqrt{65}}\right)$

3. Substitute Back: With $\theta$ known, substitute back into any other trigonometric expressions or equations to solve for $x$.

The Provided Geometric Approach

The image uses a different method, involving a right triangle and algebraic manipulations. The steps in the image are not fully complete, but they are making use of the Pythagorean theorem and solving for $x$.

Conclusion

The equation $\cos\theta + 8\sin\theta = 4$ can be solved by transforming it into a cosine function and then solving for $\theta$. The approach in the image is slightly different, involving a geometric method, which seems to ultimately solve for $x$ and then presumably $\theta$.

Do you need any further clarification or details on this?

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Related Questions:

1. How can trigonometric identities simplify solving such equations?
2. Why is it useful to rewrite the equation as $R\cos(\theta - \alpha)$?
3. What steps would you follow to graph the equation $\cos\theta + 8\sin\theta = 4$?
4. Can you explain the geometric approach used in the image in more detail?
5. How would you solve for $\theta$ if the equation was $\cos\theta + k\sin\theta = c$ with different constants?

Tip:

Always consider transforming trigonometric equations into a single trigonometric function to simplify solving for angles like $\theta$.